91Ó°ÊÓ

When working with subtraction in algebra, particularly with complex numbers, it's critical to correctly distribute the negative sign. In complex numbers, which are written in the form of \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, subtracting one number from another involves a crucial step: the distribution of the negative sign.

For example, consider the expression \( (-84+91i) - (28+72i) \). When we distribute the negative sign here, we effectively change the sign of each term in the second complex number. This leads to the new expression \( (-84+91i) + (-28-72i) \). Note the plus sign after the parenthesis, which now precedes the changed-sign second complex number. This step is foundational as it correctly sets up the equation for the next step: combining like terms.

The next step in simplifying expressions, especially when dealing with the subtraction of complex numbers, is to combine like terms. Like terms are terms that have the same variables raised to the same power, which, in the case of complex numbers, means we combine real parts with real parts and imaginary parts with imaginary parts separately.

After distributing the negative sign in our example, we have \( (-84+91i) + (-28-72i) \). To combine like terms here, we add \( -84 \) and \( -28 \) to get the real component of the resulting complex number. Similarly, we add \( 91i \) and \( -72i \) to get the imaginary component. This step simplifies the process by bringing together terms that can be added or subtracted to yield a simpler expression.

The last step in our sequence is to simplify the mathematical expression. Simplification is the process of making a mathematical expression as straightforward as possible by performing operations and reducing it to its most basic form.

In the context of subtracting complex numbers, after combining like terms, we would add the real parts and the imaginary parts obtained in the previous step to find a single complex number that represents the result. For our exercise, the simplification step for \( (-84+91i) + (-28-72i) \) would involve adding the real parts \( -84 \) and \( -28 \) to get \( -112 \) and adding the imaginary parts \( 91i \) and \( -72i \) to get \( 19i \) as the imaginary component. The simplified final expression of our original subtraction problem would therefore be \( -112 + 19i \).

Through these steps, administering the negative sign correctly, combining like terms accurately, and simplifying rigorously, students can confidently tackle complex number subtraction, arriving at a clear, concise, and correct outcome.